# §16.9 Zeros

Assume that and none of the is a nonpositive integer. Then has at most finitely many zeros if and only if the can be re-indexed for in such a way that is a nonnegative integer.

Next, assume that and that the and the quotients are all real. Then has at most finitely many real zeros.

These results are proved in Ki and Kim (2000). For further information on zeros see Hille (1929).